By successfully completing this course, you will be able to: a. Work with functions represented in a variety of ways and understand the connections among these representations. b. Understand the meaning of the derivative in terms of a rate of change and local linear approximation, and use derivatives to solve a variety of problems. c. Understand the relationship between the derivative and the definite integral. d. Communicate mathematics both orally and in well-written sentences to explain solutions to problems. e. Model a written description of a physical situation with a function, a differential equation, or an integral. f. Use technology to help solve problems, experiment, interpret results, and verify conclusions. g. Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement. h. Develop an appreciation of calculus as a coherent body of knowledge. Technology Requirement I will use a Texas Instruments 83 Plus graphing calculator in class regularly. You will want to have a graphing calculator as well. I recommend the TI-83 Plus or higher. I have a set of eleven (12) TI-83 Plus calculators and some are available for extended checkout from me. There is a companion website for the book at www.interactmath.com, which is very helpful, so internet access is needed. We will use the calculator in a variety of ways including: a. Conduct explorations. b. Graph functions within arbitrary windows. c. Solve equations numerically. d. Analyze and interpret results. e. Justify and explain results of graphs and equations. A Balanced Approach Current mathematical education emphasizes a Rule of Four. There are a variety of ways to approach and solve problems. The four branches of the problem-solving tree of mathematics are: a. Numerical analysis (where data points are known, but not an equation) b. Graphical analysis (where a graph is known, but not an equation) c. Analytic/algebraic analysis (traditional equation and variable manipulation) d. Verbal/written methods of representing problems (classic story problems as well as written justification of one s thinking in solving a problem) Below is an outline of topics along with a tentative timeline. Assessments are given at the end of each unit as well as intermittently during each unit. Semester finals are also given.
Unit 1: Prerequisites for Calculus (2 3 weeks) 1) Lines a) Slope b) Equations 2) Functions and Graphs a) Domains & Ranges b) Even & Odd Functions c) Piecewise d) Absolute Value e) Composite 3) Exponential Functions a) Growth b) Decay c) e x 4) Parametric Equations a) Relations b) Circles c) Lines 5) Functions and Logarithms a) One-to-One b) Inverses c) Logarithms and Their Properties 6) Trigonometric Functions a) Graphs b) Periodicity c) Even and Odd Trig Functions d) Transformations & Inverses Unit 2: Limits and Continuity (3 4 weeks) 1) Rates of Change and Limits a) Average speed and instantaneous speed b) Definition and properties of limits c) 1-sided limits d) 2-sided Limits e) Sandwich Theorem **A Graphical Exploration is used to investigate the Sandwich Theorem. Students graph y1 = x2, y2 = - x2, y3 = sin (1/x) in radian mode on graphing calculators. The limit as x approaches 0 of each function is explored in an attempt to see the limit as x approaches 0 of x2 * sin (1/x). This helps tie the graphical implications of the Sandwich Theorem to the analytical applications of it. 2) Limits Involving Infinity a) Asymptotic behavior (horizontal and vertical) b) End behavior models
c) Properties of limits (algebraic analysis) d) Visualizing limits (graphic analysis) 3) Continuity a) Continuity at a point b) Continuous functions c) Discontinuous functions i) Removable discontinuity (0/0 form) ii) Jump discontinuity (We look at y = int(x).) iii) Infinite discontinuity d) Intermediate Value Theorem for Continuous Functions 4) Rates of Change and Tangent Lines a) Average rate of change b) Tangent line to a curve c) Slope of a curve (algebraically and graphically) d) Normal line to a curve (algebraically and graphically) e) Instantaneous rate of change (revisited) Unit 3: The Derivative (5 6 weeks) 1) Derivative of a Function a) Definition of the derivative (difference quotient) b) Derivative at a Point c) Relationships between the graphs of f ' and f '' d) Graphing a derivative from data e) One-sided derivatives 2) Differentiability a) Cases where f '(x) might fail to exist b) Local linearity **An exploration is conducted with the calculator in table groups. Students will graph y1 = x + 1 and y2 = [(x2 + 0.0001) + 0.99]. They investigate the graphs near x = 0 by zooming in repeatedly. The students discuss the local linearity of each graph and whether each function appears to be differentiable at x = 0. c) Derivatives on the calculator (Numerical derivatives using NDERIV) d) Symmetric difference quotient e) Relationship between differentiability and continuity f) Intermediate Value Theorem for Derivatives 3) Rules for Differentiation a) Constant, Power, Sum, Difference, Product, Quotient Rules b) Higher order derivatives 4) Applications of the Derivative a) Position, velocity, acceleration, and jerk b) Particle motion c) L Hopital s Rule d) Economics
i) Marginal cost ii) Marginal revenue iii) Marginal profit 5) Derivatives of Trigonometric Functions 6) Chain Rule 7) Implicit Differentiation a) Differential method b) y method 8) Derivatives of Inverse Trigonometric Functions 9) Derivatives of Exponential and Logarithmic Functions Unit 4: Applications of the Derivative (5 6 weeks) 1) Extreme Values a) Relative Extrema b) Absolute Extrema c) Extreme Value Theorem d) Definition of a Critical Point 2) Implications of the Derivative a) Rolle s Theorem b) Mean Value Theorem c) Increasing and decreasing functions 3) Connecting f and f with the graph of f(x) a) First derivative test for relative max/min b) Second derivative i. Concavity ii. Inflection points iii. Second derivative test for relative max/min 4) Optimization problems 5) Linearization models a) Local linearization **An exploration using the graphing calculator is conducted in table groups where students graph f(x) = (x2 + 0.0001)0.25 + 0.9 around x = 0. Students algebraically find the equation of the line tangent to f(x) at x = 0. Students then repeatedly zoom in on the graph of f(x) at x = 0. Students are then asked to approximate f (0.1) using the tangent line and then calculate f (0.1) using the calculator. This is repeated for the same function, but different x values further and further away from x = 0. Students then individually write about and then discuss with their tablemates the use of the tangent line in approximating the value of the function near (and not so near) x = 0. b) Tangent line approximation c) Differentials
6) Related Rates Unit 5: The Definite Integral (3 4 weeks) 1) Approximating Areas a) Riemann sums i. Left sums ii. Right sums iii. Midpoint sums iv. Trapezoidal sums b) Definite integrals **Students are asked to graph, by hand, a constant function of their choosing. Then they are asked to calculate a definite integral from x = -3 to x = 5 using known geometric methods. Students then share their work with their tablemates and are asked to come up with a table observation. Those observations are shared with other tables and a formula is discovered. 2) Properties of Definite Integrals a) Power rule b) Mean value theorem for definite integrals **An exploration is conducted to show students the geometry of the mean value theorem for definite integrals and how it is connected to the algebra of the theorem. 3) The Fundamental Theorem of Calculus a) Part 1 b) Part 2 Unit 6: Differential Equations and Mathematical Modeling (4 weeks) 1) Slope Fields 2) Antiderivatives a) Indefinite integrals b) Power formulas c) Trigonometric formulas d) Exponential and Logarithmic formulas 3) Separable Differential Equations a) Growth and decay b) Slope fields (Resources from the Wolfram Mathematic website are liberally used.) i. General differential equations ii. Newton s law of cooling 4) Logistic Growth Unit 7: Applications of Definite Integrals (3 weeks) 1) Integral as net change a. Calculating distance traveled (particle motion)
b. Consumption over time c. Net change from data 2) Area between curves a. Area between a curve and an axis i. Integrating with respect to x ii. Integrating with respect to y b. Area between intersecting curves i. Integrating with respect to x ii. Integrating with respect to y 3) Calculating volume a. Cross sections b. Disc method c. Shell method Unit 8: Review/Test Preparation (time varies, generally 3 5 weeks) 1) Multiple-choice practice (AP Calculus AB Test on-line practice tests) a) Test taking strategies are emphasized b) Individual and group practice are both used 2) Free-response practice (AP Calculus AB Test on-line practice tests.) a) Rubrics are reviewed so students see the need for complete answers b) Students collaborate to formulate team responses c) Individually written responses are crafted. Attention to full explanations is emphasized Unit 9: After the exam 1) Projects designed to incorporate this year s learning in applied ways 2) Research projects on the historical development of mathematics with a focus on calculus 3) Advanced integration techniques 4) A look at college math requirements and expectations including placement exams Textbook: Finney, Demana, Waits, and Kennedy. Calculus: Graphical, Numerical, and Algebraic. Third edition. Pearson, Prentice Hall, 2007. This textbook will be our primary resource. You will benefit from reading it. It contains a number of interesting explorations that we will conduct with the goal that you discover fundamental calculus concepts. I will be using the on-line version of the text at www.interactmath.com in class. I believe this will be a good study tool for you to use from home as well. I encourage cooperative learning, and I believe our entire class benefits from us all working together to help one another construct understanding. My hope is that you want to learn as much as you can about calculus.